Whakaoti i te x^2-6x-3=0

$$x ^ { 2 } - 6 x - 3 = 0$$

Whakaoti mō x

$$x=2\sqrt{3}+3\approx 6.464101615$$

$$x=3-2\sqrt{3}\approx -0.464101615$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$x^{2}-6x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -6 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-3\right)}}{2}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-3\right)}}{2}$$

Whakareatia -4 ki te -3.

$$x=\frac{-\left(-6\right)±\sqrt{36+12}}{2}$$

Tāpiri 36 ki te 12.

$$x=\frac{-\left(-6\right)±\sqrt{48}}{2}$$

Tuhia te pūtakerua o te 48.

$$x=\frac{-\left(-6\right)±4\sqrt{3}}{2}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±4\sqrt{3}}{2}$$

Nā, me whakaoti te whārite x=\frac{6±4\sqrt{3}}{2} ina he tāpiri te ±. Tāpiri 6 ki te 4\sqrt{3}.

$$x=\frac{4\sqrt{3}+6}{2}$$

Whakawehe 6+4\sqrt{3} ki te 2.

$$x=2\sqrt{3}+3$$

Nā, me whakaoti te whārite x=\frac{6±4\sqrt{3}}{2} ina he tango te ±. Tango 4\sqrt{3} mai i 6.

$$x=\frac{6-4\sqrt{3}}{2}$$

Whakawehe 6-4\sqrt{3} ki te 2.

$$x=3-2\sqrt{3}$$

Kua oti te whārite te whakatau.

$$x=2\sqrt{3}+3 x=3-2\sqrt{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$x^{2}-6x-3=0$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x^{2}-6x-3-\left(-3\right)=-\left(-3\right)$$

Mā te tango i te -3 i a ia ake anō ka toe ko te 0.

$$x^{2}-6x=-\left(-3\right)$$

Tango -3 mai i 0.

$$x^{2}-6x=3$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=3+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=3+9$$

Tāpiri 3 ki te 9.

$$x^{2}-6x+9=12$$

Tauwehea x^{2}-6x+9. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=12$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{12}$$

Whakarūnātia.

$$x-3=2\sqrt{3} x-3=-2\sqrt{3}$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=2\sqrt{3}+3 x=3-2\sqrt{3}$$